In[]:=
a=1;​​Δt=0.25;​​Δx=1;​​n=10;(*no.ofpoints*)​​U=Table[u[i,1],{i,10}];​​s1=SparseArray[{Band[{1,2}]1,Band[{2,1}]-1},{10,10}];​​s2=SparseArray[{{1,1}-2,{1,2}1,{10,9}-1,{10,10}2}];​​s3=s1+s2;
In[]:=
b=SparseArray[{{1,1}Sin[4πΔti],{10,1}->0}];​​A=
1
2Δt
s1+
a
2Δx
s3;
​
A.Ub;​​Sol1=Solve[%,U]​​sol2=NDSolveValue[{D[v[t,x],t]+aD[v[t,x],x]0,v[t,0]Sin[4πt],v[0,x]0},v,{t,0.25,1},{x,1,10}];​​
Out[]=
{{u[1,1]{0.125Sin[3.14159i]},u[2,1]{0.375Sin[3.14159i]},u[3,1]{0.125Sin[3.14159i]},u[4,1]{0.375Sin[3.14159i]},u[5,1]{0.125Sin[3.14159i]},u[6,1]{0.375Sin[3.14159i]},u[7,1]{0.125Sin[3.14159i]},u[8,1]{0.375Sin[3.14159i]},u[9,1]{0.125Sin[3.14159i]},u[10,1]{0.375Sin[3.14159i]}}}